`int(x+1)/(sqrt(2x^(2)+x-3))dx`

Answer

To solve the integral

\[ I = \int \frac{x + 1}{\sqrt{2x^2 + x - 3}} \, dx, \]

we can start by rewriting the numerator \(x + 1\) in terms of the derivative of the denominator \(2x^2 + x - 3\).

### Step 1: Rewrite the Numerator
We want to express \(x + 1\) as a linear combination of the derivative of the denominator. The derivative of \(2x^2 + x - 3\) is

\[ \frac{d}{dx}(2x^2 + x - 3) = 4x + 1. \]

We can write:

\[ x + 1 = a(4x + 1) + b, \]

where \(a\) and \(b\) are constants we need to determine.

### Step 2: Set Up the Equation
Expanding the right-hand side gives:

\[ x + 1 = (4a)x + (a + b). \]

### Step 3: Compare Coefficients
Now, we compare coefficients of \(x\) and the constant term:

1. For \(x\): \(4a = 1 \implies a = \frac{1}{4}\)
2. For the constant: \(a + b = 1\)

Substituting \(a = \frac{1}{4}\) into the second equation:

\[ \frac{1}{4} + b = 1 \implies b = 1 - \frac{1}{4} = \frac{3}{4}. \]

### Step 4: Rewrite the Integral
Now we can rewrite the integral \(I\):

\[ I = \int \frac{\frac{1}{4}(4x + 1) + \frac{3}{4}}{\sqrt{2x^2 + x - 3}} \, dx. \]

This can be separated into two integrals:

\[ I = \frac{1}{4} \int \frac{4x + 1}{\sqrt{2x^2 + x - 3}} \, dx + \frac{3}{4} \int \frac{1}{\sqrt{2x^2 + x - 3}} \, dx. \]

### Step 5: Define the Integrals
Let

\[ I_1 = \int \frac{4x + 1}{\sqrt{2x^2 + x - 3}} \, dx, \]
and

\[ I_2 = \int \frac{1}{\sqrt{2x^2 + x - 3}} \, dx. \]

Thus, we have:

\[ I = \frac{1}{4} I_1 + \frac{3}{4} I_2. \]

### Step 6: Solve \(I_1\)
For \(I_1\), we can use the substitution \(t = 2x^2 + x - 3\), then \(dt = (4x + 1)dx\). This gives:

\[ I_1 = \int \frac{dt}{\sqrt{t}} = 2\sqrt{t} + C. \]

Substituting back gives:

\[ I_1 = 2\sqrt{2x^2 + x - 3}. \]

### Step 7: Solve \(I_2\)
For \(I_2\), we can also use the same substitution \(t = 2x^2 + x - 3\):

\[ I_2 = \int \frac{1}{\sqrt{t}} dt = 2\sqrt{t} + C. \]

Substituting back gives:

\[ I_2 = 2\sqrt{2x^2 + x - 3}. \]

### Step 8: Combine Results
Now substituting \(I_1\) and \(I_2\) back into the expression for \(I\):

\[ I = \frac{1}{4}(2\sqrt{2x^2 + x - 3}) + \frac{3}{4}(2\sqrt{2x^2 + x - 3}) = \frac{5}{4}\sqrt{2x^2 + x - 3} + C. \]

### Final Result
Thus, the final result for the integral is:

\[ I = \frac{5}{4}\sqrt{2x^2 + x - 3} + C. \]